1. Lock-in amplifiers

2. Signals and noise Frequency dependence of noise Low frequency ~ 1 / f
example: temperature (0.1 Hz) , pressure (1 Hz), acoustics ( Hz)
High frequency ~ constant = white noise
example: shot noise, Johnson noise, spontaneous emission noise
Total noise depends strongly on signal freq
worst at DC, best in white noise region
Problem -- most signals at DC
Total noise in 10 Hz bandwidth
Signal at DC
1/f noise
log(Vnoise)
10 Hz
White noise
kHz
log( f )
log(Vnoise)
log( f )
Noise amplitude
1/f noise
White noise
kHz
Signal at 1 kHz
1/f noise
log(Vnoise)
White noise
10 Hz
kHz
log( f )

3. Lock-in amplifiers Shift signal out to higher frequencies Approach:
Modulate signal, but not noise, at high freq
no universal technique -- art
example: optical chopper wheel, freq modulation
Detect only at modulation frequency
Noise at all other frequencies averages to zero
Use demodulator and low-pass filter

4. Demodulation / Mixing Multiply input signal by sine wave
Sum and difference freq generated
Compare to signal addition -- interference
Signal frequency close to reference freq
low freq beat
DC for equal freq sine waves
DC output level depends on relative phase
Two sine waves
Product
Sum

5. Signal freq approaches ref freq
Beat frequency approaches DC as signal freq approaches ref freq
Reference
Signal freq
vs ref freq 1
1.05
1.1
1.15
1.2
1.25
Mixer outputs

6. Phase sensitive detection
Signal freq matches reference freq
Reference = sin(2pft)
Signal = sin(2pft + f)
f is signal phase shift
Product = cos(f) - cos(2pft)
DC part
Signal
phase
shift f
0.2 p
0.4 p
0.6 p
0.8 p p
Reference wave
-- signal times reference
Product waveforms

7. Low pass filter Removes noise Example -- modulate above 1/f noise
noise slow compared to reference freq
noise converted to slowly modulated sine wave
averages out to zero over 1 cycle
Low pass filter integrates out modulated noise
leaves signal alone
Demodulated signal
After mixer
Voltage
time
After mixer & low pass
Reference
Input
Output
Mixer
Low pass
filter
Buffer
Lock-in amplifier

8. Typical LIA low pass filters
For weak signal buried in noise
Ideal low pass filter blocks all except signal
Approximate ideal filter with cascaded low pass filters
18 db/oct
12 db/oct
6 db/oct
Ideal
log
gain
frequency

9. Phase control Reference has phase control Can vary from 0 to 360°
Arbitrary input signal phase
Tune reference phase to give maximum DC output
Reference
Phase
shift f
Input
Output
Mixer

10. Reference options Option 1 -- Internal reference
System
Lock-in amplifier
Option 1 -- Internal reference
best performance
stable reference freq
Option 2 -- External reference
System generates reference
ex: chopper wheel
Lock internal ref to system ref
use phase locked loop (PLL)
source of name “lock-in amplifier”
Mixer
Signal
Reference
System
Lock-in amplifier
Mixer
Signal
Reference
VCO
PLL
Integrate

11. Analog mixer Direct multiplication Switching mixer Multiplying mixer
accurate
not enough dynamic range
weak signal buried in noise
Switching mixer
big dynamic range
but also demodulates harmonics
Multiplying mixer
Switching mixer
Harmonic content of square wave 1
1/3
1/5
1/7
1/9

12. Switching mixer design
Sample switching mixer
Back-to-back FETs
example: 1 n-channel & 1 p-channel
feed signal to one FET, inverted signal to second FET
Apply square wave to gates
upper FET conducts on positive part of square wave
lower FET conducts on negative part
Switching mixer circuit
n-channel FET p n
Signal
voltage
source
drain
gate
bias

13. Signals with harmonic content
Option 1: Use multi-switch mixer
approximate sine wave
cancel out first few harmonic signals
Option 2: Filter harmonic content from signal
bandpass filter at input
Q > 100
Lock-in amp with input filter

14. Digital mixers Digitize input with DAC Multiply in processor
Advantages:
Accurate sine wave multiplication
No DC drift in low pass filters
Digital signal enhancement
Problems:
Need 32 bit DAC for signals buried in noise
Cannot digitize 32 bits at 100 kHz rates
Should be excellent for slow servos
Ex: tele-medicine, temperature controllers
Digital processing can compensate for certain system time delays ?

15. Lock-in amps in servos F(x) x Lock to resonance peak
Servos only lock to zero
Need to turn peak into zero
Take derivative of lineshape
modulate x-voltage
F(x)-voltage amplitude like derivative
Use lock-in amp to extract amplitude of F(x)
“DC” part of mixer output
filter with integrator, not low-pass
Take derivative with lock-in x
F(x)
No fundamental
only 2 f signal

16. Lock-in amps for derivative
Lock-in turns sine wave signal into DC voltage
At peak of resonance
no signal at modulation freq
lock-in output crosses zero
Discriminant
use to lock
Input signal
F(x) x
Lock-in
output
(derivative)
Zero crossing
at resonance

17. Effect of modulation on lineshape
Start with resonance lineshape
Intensity vs PZT voltage: I = I0 exp( -V2)
Modulate voltage: V= V0 sin (2 p f t)
Modified lineshape
Analog to numerical derivatives
Derivative is: I’ = I(V+ DV) - I(V) / DV
Set DV = 1
Modulation replaces DV= V0 sin (2 p f t)
Derivative is sine wave part
Assumes is V0 small V V t I t

18. Effect of modulation amplitude
For large modulation amps
Distortion and broadening
Modulation like a noise source
Always use minimum necessary
Modulation amplitude
0.05 linewidth
0.1
0.2
0.5 linewidth 1 2
Expanded scan

19. Mixer outputs Maximum mixer output modulation ~ 1 linewidth
Modulation amplitude
0.1 linewidth
0.2
Maximum mixer output
modulation ~ 1 linewidth
saturates and broadens
0.5 linewidth 1 2
Mixer out
0.1 linewidth
0.2
0.5 1 2

20. Fabry-Perot servo
Lock to peak transmission of high Q Fabry-Perot etalon
Use lock-in amp to give discriminant
No input bandpass -- or low Q < 2
Bandpass rolloff usually 2-pole or greater
No low pass filter -- replace with integrator
Low pass filter removes noise
Need noise to produce correction
Design tips
reference freq must exceed servo bandwidth by factor of ~ 10
but PZT bandwidth is servo limiter
use PZT resonance for modulation
Acoustic noise
Laser
Fabry-Perot PD
LIA
Sum
& HV
reference